Two-dimensional collisions happen when objects collide and move away in directions that are not all along one line. They are common in billiards, air hockey, particle physics, and vehicle crash analysis. The key idea is that momentum is a vector, so both size and direction matter.
Breaking each velocity into x and y components makes a complicated glancing collision easier to analyze.
In a low-friction collision, the total momentum of the colliding objects is conserved separately in the horizontal and vertical directions. This means the total x-momentum before equals the total x-momentum after, and the total y-momentum before equals the total y-momentum after. If the collision is elastic, kinetic energy is also conserved, but momentum conservation applies to both elastic and inelastic collisions when outside forces are negligible.
A worked diagram usually shows incoming and outgoing velocity vectors, then resolves each vector into components to build two conservation equations.
Key Facts
- Momentum is a vector: p = mv.
- Horizontal momentum conservation: m1v1x,i + m2v2x,i = m1v1x,f + m2v2x,f.
- Vertical momentum conservation: m1v1y,i + m2v2y,i = m1v1y,f + m2v2y,f.
- Velocity components are vx = v cos(theta) and vy = v sin(theta), when theta is measured from the positive x-axis.
- For an elastic collision, kinetic energy is conserved: 1/2 m1v1i^2 + 1/2 m2v2i^2 = 1/2 m1v1f^2 + 1/2 m2v2f^2.
- For a perfectly inelastic collision, objects stick together and share a final velocity, but total momentum is still conserved.
Vocabulary
- Momentum
- Momentum is the product of an object's mass and velocity, and it points in the same direction as the velocity.
- Vector component
- A vector component is the part of a vector that lies along a chosen axis, such as the x-direction or y-direction.
- Glancing collision
- A glancing collision is an off-center collision in which objects move away at angles after impact.
- Elastic collision
- An elastic collision is a collision in which total momentum and total kinetic energy are both conserved.
- Inelastic collision
- An inelastic collision is a collision in which total momentum is conserved but kinetic energy is not fully conserved.
Common Mistakes to Avoid
- Adding speeds instead of vector components is wrong because momentum must be conserved direction by direction, not just by using speed values.
- Forgetting signs on components is wrong because motion to the left or downward usually has negative momentum in a chosen coordinate system.
- Assuming kinetic energy is always conserved is wrong because only elastic collisions conserve kinetic energy, while inelastic collisions convert some kinetic energy into heat, sound, or deformation.
- Using degrees from the wrong axis is wrong because vx = v cos(theta) and vy = v sin(theta) only apply directly when theta is measured from the positive x-axis.
Practice Questions
- 1 A 0.20 kg puck moves at 5.0 m/s along the positive x-axis and strikes a 0.30 kg puck initially at rest. After the collision, the 0.20 kg puck moves at 3.0 m/s at 30 degrees above the x-axis. What are the x and y components of the 0.30 kg puck's final momentum?
- 2 A 0.50 kg ball traveling at 4.0 m/s at 20 degrees above the x-axis collides with a 0.50 kg ball initially at rest. Afterward, the first ball moves at 2.5 m/s at 40 degrees above the x-axis. Use momentum components to find the final velocity components of the second ball.
- 3 In a glancing collision on a nearly frictionless table, why can the total x-momentum stay constant even though the x-momentum of each individual puck changes?